Braided Groups and Quantum Fourier Transform

✍ Scribed by V. Lyubashenko; S. Majid

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 908 KB
• Volume: 166
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1994.1165

✦ Synopsis

We show that acting on every finite-dimensional factorizable ribbon Hopf algebra (H) there are invertible operators (\mathscr{S}{-}, \mathscr{T}) obeying the modular identities (\left(\mathscr{S}{-} \mathscr{T}\right)^{3}=\lambda \mathscr{P}^{2}), where (\lambda) is a constant. The class includes the finite-dimensional quantum groups (u_{q}(g)) associated to complex simple Lie algebras. We give the example of (u_{q}(s l(2))) at a root of unity in detail, as well as an example relating to anyons. The operator (\mathscr{S}_{-})plays the role of "quantum Fourier Transform" and acts naturally on (H) viewed by transmutation as a braided group (H) (a braided-cocommutative Hopf algebra in a braided category). It obeys (\mathscr{S}^{2}=\underline{s}^{-1}), where (s) is the antipode of (\underline{H}). The results follow as an application of previous category-theoretical constructions. (\bar{S} 1994) Academic Press, Inc.